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  <div class="section" id="numpy-linalg-eigh">
<h1>numpy.linalg.eigh<a class="headerlink" href="#numpy-linalg-eigh" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.linalg.eigh">
<code class="sig-prename descclassname">numpy.linalg.</code><code class="sig-name descname">eigh</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">UPLO='L'</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/linalg/linalg.py#L1314-L1449"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.linalg.eigh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the eigenvalues and eigenvectors of a complex Hermitian
(conjugate symmetric) or a real symmetric matrix.</p>
<p>Returns two objects, a 1-D array containing the eigenvalues of <em class="xref py py-obj">a</em>, and
a 2-D square array or matrix (depending on the input type) of the
corresponding eigenvectors (in columns).</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>a</strong><span class="classifier">(…, M, M) array</span></dt><dd><p>Hermitian or real symmetric matrices whose eigenvalues and
eigenvectors are to be computed.</p>
</dd>
<dt><strong>UPLO</strong><span class="classifier">{‘L’, ‘U’}, optional</span></dt><dd><p>Specifies whether the calculation is done with the lower triangular
part of <em class="xref py py-obj">a</em> (‘L’, default) or the upper triangular part (‘U’).
Irrespective of this value only the real parts of the diagonal will
be considered in the computation to preserve the notion of a Hermitian
matrix. It therefore follows that the imaginary part of the diagonal
will always be treated as zero.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>w</strong><span class="classifier">(…, M) ndarray</span></dt><dd><p>The eigenvalues in ascending order, each repeated according to
its multiplicity.</p>
</dd>
<dt><strong>v</strong><span class="classifier">{(…, M, M) ndarray, (…, M, M) matrix}</span></dt><dd><p>The column <code class="docutils literal notranslate"><span class="pre">v[:,</span> <span class="pre">i]</span></code> is the normalized eigenvector corresponding
to the eigenvalue <code class="docutils literal notranslate"><span class="pre">w[i]</span></code>.  Will return a matrix object if <em class="xref py py-obj">a</em> is
a matrix object.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>LinAlgError</strong></dt><dd><p>If the eigenvalue computation does not converge.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.linalg.eigvalsh.html#numpy.linalg.eigvalsh" title="numpy.linalg.eigvalsh"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eigvalsh</span></code></a></dt><dd><p>eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.</p>
</dd>
<dt><a class="reference internal" href="numpy.linalg.eig.html#numpy.linalg.eig" title="numpy.linalg.eig"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eig</span></code></a></dt><dd><p>eigenvalues and right eigenvectors for non-symmetric arrays.</p>
</dd>
<dt><a class="reference internal" href="numpy.linalg.eigvals.html#numpy.linalg.eigvals" title="numpy.linalg.eigvals"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eigvals</span></code></a></dt><dd><p>eigenvalues of non-symmetric arrays.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.8.0.</span></p>
</div>
<p>Broadcasting rules apply, see the <a class="reference internal" href="../routines.linalg.html#module-numpy.linalg" title="numpy.linalg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.linalg</span></code></a> documentation for
details.</p>
<p>The eigenvalues/eigenvectors are computed using LAPACK routines <code class="docutils literal notranslate"><span class="pre">_syevd</span></code>,
<code class="docutils literal notranslate"><span class="pre">_heevd</span></code>.</p>
<p>The eigenvalues of real symmetric or complex Hermitian matrices are
always real. <a class="reference internal" href="#rc702e98a756a-1" id="id1">[1]</a> The array <em class="xref py py-obj">v</em> of (column) eigenvectors is unitary
and <em class="xref py py-obj">a</em>, <em class="xref py py-obj">w</em>, and <em class="xref py py-obj">v</em> satisfy the equations
<code class="docutils literal notranslate"><span class="pre">dot(a,</span> <span class="pre">v[:,</span> <span class="pre">i])</span> <span class="pre">=</span> <span class="pre">w[i]</span> <span class="pre">*</span> <span class="pre">v[:,</span> <span class="pre">i]</span></code>.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="rc702e98a756a-1"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>G. Strang, <em>Linear Algebra and Its Applications</em>, 2nd Ed., Orlando,
FL, Academic Press, Inc., 1980, pg. 222.</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">linalg</span> <span class="k">as</span> <span class="n">LA</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
<span class="go">array([[ 1.+0.j, -0.-2.j],</span>
<span class="go">       [ 0.+2.j,  5.+0.j]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">LA</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="p">;</span> <span class="n">v</span>
<span class="go">array([0.17157288, 5.82842712])</span>
<span class="go">array([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary</span>
<span class="go">       [ 0.        +0.38268343j,  0.        -0.92387953j]])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="c1"># verify 1st e-val/vec pair</span>
<span class="go">array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">])</span> <span class="o">-</span> <span class="n">w</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="c1"># verify 2nd e-val/vec pair</span>
<span class="go">array([0.+0.j, 0.+0.j])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="c1"># what happens if input is a matrix object</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
<span class="go">matrix([[ 1.+0.j, -0.-2.j],</span>
<span class="go">        [ 0.+2.j,  5.+0.j]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">LA</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="p">;</span> <span class="n">v</span>
<span class="go">array([0.17157288, 5.82842712])</span>
<span class="go">matrix([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary</span>
<span class="go">        [ 0.        +0.38268343j,  0.        -0.92387953j]])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># demonstrate the treatment of the imaginary part of the diagonal</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">5</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mi">9</span><span class="o">-</span><span class="mi">2</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="n">j</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
<span class="go">array([[5.+2.j, 9.-2.j],</span>
<span class="go">       [0.+2.j, 2.-1.j]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># with UPLO=&#39;L&#39; this is numerically equivalent to using LA.eig() with:</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">5.</span><span class="o">+</span><span class="mf">0.</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.</span><span class="o">-</span><span class="mf">2.</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.</span><span class="o">+</span><span class="mf">2.</span><span class="n">j</span><span class="p">,</span> <span class="mf">2.</span><span class="o">-</span><span class="mf">0.</span><span class="n">j</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
<span class="go">array([[5.+0.j, 0.-2.j],</span>
<span class="go">       [0.+2.j, 2.+0.j]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">wa</span><span class="p">,</span> <span class="n">va</span> <span class="o">=</span> <span class="n">LA</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">wb</span><span class="p">,</span> <span class="n">vb</span> <span class="o">=</span> <span class="n">LA</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">wa</span><span class="p">;</span> <span class="n">wb</span>
<span class="go">array([1., 6.])</span>
<span class="go">array([6.+0.j, 1.+0.j])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">va</span><span class="p">;</span> <span class="n">vb</span>
<span class="go">array([[-0.4472136 +0.j        , -0.89442719+0.j        ], # may vary</span>
<span class="go">       [ 0.        +0.89442719j,  0.        -0.4472136j ]])</span>
<span class="go">array([[ 0.89442719+0.j       , -0.        +0.4472136j],</span>
<span class="go">       [-0.        +0.4472136j,  0.89442719+0.j       ]])</span>
</pre></div>
</div>
</dd></dl>

</div>


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